This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here's the solution to question 22. Question 22: a) (i) Represent the number of items bought by Juma, Tom, and Mary using a 3 × 2 matrix. Juma bought 3 dozens of Maize flour, which is 3 × 12 = 36 packets. The items are Maize flour (packets) and Sugar (kg). Quantity Matrix (Q) = 36 & 20 \\ 45 & 18 \\ 38 & 25 Where rows represent Juma, Tom, Mary respectively, and columns represent Maize flour, Sugar respectively. (ii) Represent the prices of items bought using a 2 × 1 matrix. The price of a packet of maize flour is Ksh 170. The price of a kilogram of sugar is Ksh 210. Price Matrix (P) = 170 \\ 210 Where rows represent Maize flour, Sugar respectively. b) Use the matrices in (a) above to determine the total expenditure incurred by each trader and hence determine the maximum difference between the traders. Step 1: Calculate the total expenditure for each trader by multiplying the quantity matrix by the price matrix. Expenditure Matrix (E) = Q × P = 36 & 20 \\ 45 & 18 \\ 38 & 25 170 \\ 210 Step 2: Perform the matrix multiplication. For Juma: (36 × 170) + (20 × 210) = 6120 + 4200 = 10320 Ksh For Tom: (45 × 170) + (18 × 210) = 7650 + 3780 = 11430 Ksh For Mary: (38 × 170) + (25 × 210) = 6460 + 5250 = 11710 Ksh E = 10320 \\ 11430 \\ 11710 The total expenditure incurred by each trader is: Juma: Ksh 10320 Tom: Ksh 11430 Mary: Ksh 11710 Step 3: Determine the maximum difference between the traders. Difference between Tom and Juma = 11430 - 10320 = 1110 Ksh Difference between Mary and Juma = 11710 - 10320 = 1390 Ksh Difference between Mary and Tom = 11710 - 11430 = 280 Ksh The maximum difference between the traders is Ksh 1390. c) Later in the week the cost of maize flour decreased by 10% while the price of sugar increased by 20%. Juma and Mary bought the same number of items as they did earlier in the week. Use matrix method to find the expenditure of each trader. Step 1: Calculate the new prices. New price of maize flour = 170 × (1 - 0.10) = 170 × 0.90 = 153 Ksh New price of sugar = 210 × (1 + 0.20) = 210 × 1.20 = 252 Ksh Step 2: Create the new price matrix. New Price Matrix (P_new) = 153 \\ 252 Step 3: Calculate the new expenditure for each trader using the original quantity matrix and the new price matrix. New Expenditure Matrix (E_new) = Q × P_new = 36 & 20 \\ 45 & 18 \\ 38 & 25 153 \\ 252 Step 4: Perform the matrix multiplication. For Juma: (36 × 153) + (20 × 252) = 5508 + 5040 = 10548 Ksh For Tom: (45 × 153) + (18 × 252) = 6885 + 4536 = 11421 Ksh For Mary: (38 × 153) + (25 × 252) = 5814 + 6300 = 12114 Ksh The expenditure of each trader with the new prices is: Juma: Ksh 10548 Tom: Ksh 11421 Mary: Ksh 12114 3 done, 2 left today. You're making progress.